Hypergraph Saturation Irregularities

TL;DR

This paper constructs a specific family of hypergraphs demonstrating that the saturation number's normalized value does not converge as the number of vertices grows, answering a question posed by Pikhurko.

Contribution

It proves the existence of a finite hypergraph family for which the saturation number's ratio to n^{r-1} does not tend to a limit, resolving an open problem.

Findings

Existence of a hypergraph family with non-converging saturation ratio

Answer to Pikhurko's question on saturation numbers

Contribution to extremal hypergraph theory

Abstract

Let $\mathcal{F}$ be a family of $r$-graphs. An $r$-graph $G$ is called $\mathcal{F}$-saturated if it does not contain any members of $\mathcal{F}$ but adding any edge creates a copy of some $r$-graph in $\mathcal{F}$. The saturation number $\operatorname{sat}(\mathcal{F},n)$ is the minimum number of edges in an $\mathcal{F}$-saturated graph on $n$ vertices. We prove that there exists a finite family $\mathcal{F}$ such that $\operatorname{sat}(\mathcal{F},n) / n^{r-1}$ does not tend to a limit. This settles a question of Pikhurko.